L(s) = 1 | + (2.5 − 0.866i)7-s + 1.73i·13-s − 8.66i·19-s − 5·25-s − 10.3i·31-s + 37-s + 8·43-s + (5.5 − 4.33i)49-s + 8.66i·61-s − 11·67-s − 1.73i·73-s + 13·79-s + (1.49 + 4.33i)91-s − 19.0i·97-s − 19.0i·103-s + ⋯ |
L(s) = 1 | + (0.944 − 0.327i)7-s + 0.480i·13-s − 1.98i·19-s − 25-s − 1.86i·31-s + 0.164·37-s + 1.21·43-s + (0.785 − 0.618i)49-s + 1.10i·61-s − 1.34·67-s − 0.202i·73-s + 1.46·79-s + (0.157 + 0.453i)91-s − 1.93i·97-s − 1.87i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.800930228\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.800930228\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 8.66iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8.66iT - 61T^{2} \) |
| 67 | \( 1 + 11T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.632135251851938984692555237765, −7.67101250455019388651870754861, −7.28539439181109589120339752562, −6.29596164257119100121439676028, −5.48237915468850255873683290519, −4.53691187361494717572506307573, −4.09572157708754429155199017091, −2.74781668668536905373068115009, −1.89531933809183553671021224480, −0.59279181459941608195641281408,
1.27094050615740478147707588121, 2.14429156638193104562435287295, 3.33158839020362553575561736016, 4.15602799619718188719085961606, 5.13976238536448664891883172980, 5.72060992980244575548564254827, 6.52636116670720468278558279275, 7.70943260864157341491705331253, 7.953148327174725682167949539983, 8.785551217603960942057289048301